Law Enforcement is the only career where advanced education can prevent you from getting a start. Being "too intelligent" is something that can disqualify you from a job in law enforcement.
The FBI won't hire you without some previous law enforcement experience and a law degree. The INS isn't even called that anymore and they likewise are not going to hire at entry level either.
All law enforcement careers start at the bottom of the barrel. There is no such thing as advanced education resulting in starting higher up in the food chain. This is an error many young people make and they delay their entry into the job market. Police work is generally a blue collar occupation.
There simply is no way to advance in law enforcement other than through the lower ranks. However, to advance in rank, you'll need that education.
You won't need to waste any more of your youth putting off your career while you work on a masters degree. Start applying in places where you really want to work as an officer and start building that foundation immediately. Complete your masters part-time after you start working - there's no hurry.
You have to be a person of integrity. Such a quality is rare and in high demand. Your level of education if combined with a high level of integrity will get you that first job right now. Do everything possible to be in your best physical condition.
You might want to consider joining the armed forces as a reserve officer and gain a little life experience too.
Literally nothing you learned in college as a Criminal Justice major has any practical application in actual street law enforcement. It's all just theory and background information and you'll be trained from zero all over again at the police academy.
The key to achieving your goal is to first have a goal -- a specific goal. If you want to be an FBI agent, look at their qualifications and work toward that. If you want to be a city cop, start applying -- the job market is wide open -- but only for those people of integrity I mentioned.
To get what you want, you must first know what you want. Decide that first and work toward that goal and your problems will easily resolve themselves.
Good luck.